Euler and GravityDecember 2009 – A guest column by Dominic Klyve

The popular myth of the discovery of gravity goes something like this: one day, anapple fell on the head of a young Isaac Newton. After pondering this event, Newtonwrote down an equation describing an invisible force, which he called gravity. Thisequation united ideas about the paths of cannon balls and apples (terrestrial motion)with the paths of moons and planets (celestial motion). Once it was written down, itelegantly and easily explained the motion of all the planets and moons, and remainedunquestioned, revered, and perfect for centuries (at least until Einstein).

Readers who are familiar with the history of science know better; nothing is everso simple. In fact, Newton’s theory was accepted, tested, clung to, retested, ques-tioned, revised and un-revised, and finally accepted again during the 75 years afterthe publication of Newton’s Principia. More importantly for our purposes, Euler wasat the center of all of it.

Prelude to a crisis

De Causa Gravitatis

Euler had been thinking about gravity even before the worst of the chaos mentionedabove. In 1743, he published an anonymous essay, De Causa Gravitatis [Euler 1743].The essay itself is remarkable for two reasons: first, it was published anonymously.This seems to have been a technique that Euler used to get ideas out to the publicwithout having to publicly defend them, which gave him greater leeway to experimentwith bold claims. Furthermore, it was published only two years after he arrived athis new job in Berlin, and written even closer to the time of his move, and we mightguess that Euler was especially keen not to publish anything controversial before hehad settled in. The second remarkable thing about this paper is that it does not havean Enestrom number. Then man who cataloged all of Euler’s works early in the 20thcentury missed this one completely. We only know today that Euler is the author

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because Euler admitted it in a letter written 22 years later to Georges-Louis Lesage(see [Kleinert 1996] for an interesting introduction to de Causa and its discovery).

In this paper Euler examines the standard Newtonian view that gravity is a fun-damental property of all bodies, and can’t be derived from any more basic principle.Euler is not convinced that this is true, and gives an example of how an inversesquare law might be the result of some other property of the universe. Euler takes asa given that the universe is permeated by the ether, a subtle fluid existing everywherethat permits light (among other things) to travel the distances between astronomicalbodies. Next, since Bernoulli’s principle holds that fluid moving quickly relative to abody has lower density than fluid that is stationary, it may be reasonable to assumethat the density of the ether near the Earth is less than that far away from the Earth.

If we further assume that the pressure drops off with the reciprocal of the distanceto the Earth, we find something interesting. Euler writes:

“Let the absolute compression of the ether (whenit is not lessened) be denoted c. And let the dis-tance from the center of the Earth be equal to c mi-nus a quantity reciprocally proportional to x. And solet a compression of the ether in distance x from thecenter of the Earth be C = c − cg/x. From this,if body AABB is in position around the Earth, thesurface of it above AA will be pressed downwards byforce = c − cg/CA, moreover the surface below BBwill be pressed upwards by a force c − cg/CB, whichforce, since it is less than before, the body will bepressed downwards by force = cg(1/CA − 1/CB) =cg.AB/(ACBC).

And so if the magnitude of body AB is in-comprehensibly less than distance CB, we will haveAC = BC, from this the gravity of such a body inwhatever distance from the center will be as 1/AC2;this is reciprocally as a square of the distance fromthe center.” (translation by DeSchmidt and Klyve[Euler 1743])

This is, frankly, rather bizarre. Euler here suggests that the thing that holds usto the ground is a pushing force from the (comparatively) dense ether above us. Weshould note here that there’s no evidence that Euler took this explanation seriously(probably the reason for his anonymous publication in the first place). Rather, heseems to want to use this simple explanation as a reminder that we shouldn’t stoplooking for a “first cause” of gravity, and that we may be able to derive the inversesquare law from an earlier principle.

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The gravity crisis

A few years after the publication of De Causa Gravitatis, Newton’s inverse squarelaw met with challenges considerably greater than those provided by philosophicalspeculations. New developments in observational technology allowed astronomers tomeasure the location of the moon and planets to a precision not before achieved.There was just one problem – the new accurate measurements failed to conform tothose predicted by the theory. In fact, there were three significant problems with thepredictions of Newton’s laws, and three attempts to fix them. We turn next to these.

Problem 1: Lunar apsides

It was well known that Newton’s laws implied that bodies orbiting Earth would movein an ellipse. When carefully applied, in fact, Newton’s laws also show that this ellipsewill itself slowly revolve around the Earth, as a consequence of the gravitational forceof the sun. The pattern of revolution of the “lunar apsis” is complicated, but averagesabout 3◦ per month (see Figure 2), a fact well established through observations inthe early eighteenth century.

Figure 2: Precession of the apsidal line – exaggerated(figure courtesy of Rob Bradley)

The problem with this was that the revolution calculated using Newton’s inverse-square law for gravitation did not give a figure of 3◦ per month; it gave only half that.The three leading mathematicians of the time – Jean d’Alembert, Alexis Clairaut, andEuler – had attempted the calculation, and each of them came inescapably to the sameconclusion that Newton himself had in the Principia: that Newton’s laws suggesteda monthly apsis revolution of only 1.5◦. What was going on? (For an engaging andmore detailed account of the issue of the lunar apsis, readers are encouraged to readRob Bradley’s article [Bradley 2007] on this subject.)

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Problem 2: Jupiter and Saturn

The interactions of Jupiter and Saturn are devilishly complicated. The basic idea isthat when Jupiter starts to catch up to Saturn, it pulls “backward” on Saturn, causingthe ringed planet to lose kinetic energy. Saturn then falls to a closer orbit and, a bitsurprisingly, speeds up (as a consequence of Kepler’s third law). When Jupiter passesSaturn, it pulls “forward”, giving Saturn more kinetic energy, moving it to a fartherorbit, and slowing it down. The mathematics of all of this is made difficult by theplanets’ elliptical orbits (see Wilson [Wilson 2007] for details), but it’s possible towork out, using calculus, just what the change in Jupiter’s and Saturn’s orbits shouldbe. As was the case with the moon, however, the theoretically calculated differencedid not match the observed difference.

Problem 3: The eclipse of 1748

The first two problems described above were very public problems – all of the majorplayers of the day were aware of them, and historians of science have often discussedthem. Another issue that arose in the context of 18th century mechanics, however, isless well known, and was an issue only for Euler.

In 1746, when thirty-nine years old, he published the Opuscula varii argumenti[E80], a fascinating book containing eight essays on such topics as light and colors, thematerial basis for thought, studies on “the nature of the smallest parts of matter,” anda discourse describing the frictional force the planets feel as a result of passing throughthe ether. Also included was his new collection of tables describing the motion of themoon. Deriving the motion as carefully as he could using the calculus, he refinedand calibrated the results using observations [Wilson 2007]. These tables became themost accurate available, predicting the location of the moon to an accuracy of 1/2arc minute.

Flushed with the success of his work, he anxiously awaited a solar eclipse which hehad calculated to occur on July 25, 1748. When the eclipse finally happened, Eulerfound he had correctly predicted the length and the start time of the eclipse, but thathe had gotten the length of the “annular” phase (when the sun can be seen as a ringaround the moon) completely wrong. Euler wrote “reflections” about this agreementand disagreement during the weeks following the eclipse [E117], and a more detailedpaper trying to explain the disagreement a few months later [E141]. In each of thesepapers he tries, but never fully succeeds, to account for his error, and he must havebegun to wonder: is the problem with his calculation the same problem that everyonewas finding with the moon, Jupiter, and Saturn?

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Desperate for a Solution

At the end of the 1740’s, Euler, D’Alembert, and Clairaut, were rapidly running outof ideas. If their mathematics, together with Newton’s law of gravity, failed to explainthe motion of bodies in the cosmos, something must be wrong. And failing to findany errors in their calculations, they reluctantly turned elsewhere.

Attempted Solution 1: Questioning the inverse square law

One of the boldest attempts to reconcile the observed and theoretical descriptions ofthe moon’s motion was made not by Euler, but Clairaut, who announced in November1747 at a public session in the French Academy of Sciences that Newton’s theory ofgravity was wrong. That one of the leading mathematicians in the world wouldpublicly make such a claim is evidence of just how dire the situation had become.Clairaut suggested that the strength of gravity was proportional not to 1

r2 , but themore complicated

1

r2+

c

r4

for some constant c. Over large distances, the c/r4 term would effectively disappear,accounting for the utility of the inverse square law over large distances. He then begantrying to find a value of c which could account for the moon’s motion. He wouldcontinue to pursue this idea until May 17, 1749, when he made an equally dramaticannouncement in which he claimed that Newton was right after all. [Hankins 1985]

Attempted Solution 2: Changing the shape of the moon

During the period when Clairaut was entertaining the change to Newton’s law, Eulercontinued to try to explain the motion of the moon without accepting that Newtonmay have been wrong. In a surprising letter of January 20, 1748, D’Alembert wroteto Euler [Euler 1980] to suggest a new theory: perhaps the moon (or at least itsdistribution of mass) was not spherical. If, after all, we only see one side from theEarth, we can’t know how far back it truly extends. And perhaps if it extends farenough back, the apsidal motion would indeed be 3◦, as observed. In an even moresurprising response written less than four weeks later [Euler 1980], Euler says that hetoo had considered this idea, and had worked out the details! He found that moonwould have to extend back about 21

2Earth diameters in the direction away from us,

which seemed untenable (translations and more discussion of this correspondence canbe found in Bradley [Bradley 2007]).

Attempted Solution 3: Moving Berlin

A third attempt to fix the discrepancy between prediction and measurements wasemployed by Euler in his second paper on the solar eclipse. In E141, Euler tries to

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explain the failure of this predictions about the eclipse by writing

“Yet, toward correcting the error in the duration of the annulus, I mustnote that in my calculations I had assumed the latitude of Berlin was52◦, 36′; now in fact the last observations that Mr. Kies made with theexcellent Quadrant that Mr. de Maupertuis gave to the Academy onlygave its elevation to be 52◦, 31′, 30′′, so I had placed Berlin too far Northby 4′, 30′′. Just glancing at the chart of this Eclipse published at Nurnbergoffers the assurance that if Berlin were situated 4′ more to the north, theduration of the annulus would have considerably lengthened and wouldhave been very close to my calculation.” [E141]

In other words, like lost drivers around the world, Euler blames his maps. Todayis it easy, using a software program like Starry Night, to recreate the eclipse, and lookat the effect of moving about 4′ north from Berlin. A arc minute on the surface ofthe Earth roughly corresponds to a mile, and moving four miles can be seen to haveessentially no effect. That is, the discrepancy in Euler’s maps cannot be the problem.Euler himself doesn’t redo his calculations with the new value, but merely reworksthem using a different method, and – hindsight really is easier than foresight – getsthe right answer. In the end, though, this is no answer at all. Euler’s calculationsstill fail to predict the motion of the moon.

A Solution at last

When Clairaut announced in 1749 that Newton’s inverse-square law was right afterall, Euler was anxious to know his solution. He could find no errors in his own cal-culations, and knew that either Clairaut had erred, or else had found a dramaticnew method of calculation. In order to see Clairaut’s work, Euler arranged for theannual prize of the Saint Petersburg Academy to be given to the paper which couldbest “demonstrate whether all the inequalities observed in lunar motion are in ac-cordance with Newtonian theory” (quoted in [Wilson 2007]). Euler was on the prizecommittee, and copies of all the submissions were sent to him. One can imagine himtearing open the envelope with the “anonymous” submissions, shuffling through themto find Clairaut’s (Euler would have recognized his style immediately), and eagerlyreading it. After satisfying himself of the correctness of Clairaut’s solution, he wrotetwo long and glowing letters to the French mathematician, praising his work andcongratulating him on his success.

What, then, had Euler been doing wrong? The problem turned out to be that he(and Clairaut and d’Alembert) had failed to take into account second-order effects.One way the calculus of the time was done used differentials. Given a variable x, itsvalue an infinitesimal moment later was written x+dx. If y = x2, then a small change

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in x would change y’s value to

y + dy = (x + dx)2 = x2 + 2xdx + (dx)2.

Since dx is infinitesimal, (dx)2 is really infinitesimal, and was routinely treated as0 and eliminated from future calculations (indeed, it is only the coefficient of thedx term that today we call the derivative of y). It turned out, though, that for3-body systems (Earth-Moon-Sun or Sun-Jupiter-Saturn), second-order effects arevery important. When Clairaut redid his calculations, attempting the tedious taskof keeping all the second-order terms, it turned out that Newton’s laws preciselymatched the observed behavior of the moon, Jupiter, and Saturn.

Back to Philosophy

Letters to a German Princess

By 1750, then, the perceived problems with the law of gravity had been resolved, andEuler could turn his attention back to more philosophical questions. Mathematiciansfamiliar with Euler’s work only through today’s textbooks are often surprised todiscover that Euler was also a writer of “popular science” books – a sort of 18thCentury Isaac Asimov or Stephen Jay Gould. His Letters to a German Princess[E343] contain explanations of scores of topics spanning the science of the day, fromlogic to optics, from music to the origin of evil. The letters are available online (inEnglish translation) and are well-worth reading even today. In letters 45-53 (amongothers), Euler treats gravity.

It is in this work that we see a more mature Euler. He wrote these letters in 1760,when the problems that beset Newton’s theory in the 1740’s had long since beenresolved. Nevertheless, he still shows some of the reservations he had twenty-fiveyears earlier with accepting gravity as a first principle. Indeed, in letter 46 he writesthat “Philosophers have warmly disputed, whether there actually exists a power whichacts in an invisible manner upon bodies: or whether it be an internal quality inherentin the very nature of the bodies...”

Euler’s work is meant to be read by a lay audience, and he gives several visualimages for the reader to think about. In one wonderful image, Euler writes: “Thereis a cellar under my apartment, but the floor supports me, and preserves me fromfalling into it. Were the floor suddenly to crumble away, and the arch of the cellarto tumble in at the same time, I must infallibly by precipitated into it, because mybody is heavy, like all other bodies with which we are acquainted.”

Little shows Euler to be product of his time more than his explanation of thislast qualifier: “I say, with which we are acquainted, for there may, perhaps, be bodiesdestitute of weight: such as, possibly, light itself, the elementary fire, the electric fluid,or that of the magnet: or such as the bodies of angels which have formerly appeared

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to men.” At the time, light, fire, electricity, and magnetism all were thought bymany people to consist of weightless (imponderous) bodies which diffused throughother matter, granting it properties like heat or electricity. Euler’s claim that angelsare weightless (and, for that matter, that they have “appeared to men”) was lessgrounded by scientific evidence.

Euler’s later work

Euler would continue to work on problems concerning gravity until the end of his life,even famously working on the orbit of Uranus on the day he died. In 1772 he pub-lished a third Lunar Theory, in which he used his new understanding of gravitationalcalculations to find more accurate descriptions of the motion of the moon. He neverdid fully satisfy his own curiosity about the cause of gravity but since the scientificcommunity had to wait until Einstein to get an explanation better than Newton’s, hecan perhaps be forgiven this failure. Had he somehow lived to the 20th century, how-ever, we can be sure that Euler would have found fertile ground for further researchin our modern views on gravity as well.

References

[E80] Euler, Leonhard, Opuscula varii argumenti, Berlin (1746). The lunar tablesare republished in Opera Omnia: Series 2, Volume 23, pp. 1–10. Original scansavailable at www.eulerarchive.org.

[E117] Euler, Leonhard, “Reflexions sur la derniere eclipse du Soliel du 25 jullieta. 1748.” Memoires de l’academie des sciences de Berlin 3, 1749, pp. 250–273.Reprinted in Opera Omnia: Series 2, Volume 30, pp. 51–72. The original paper isalso available at www.eulerarchive.org.

[E141] Euler, Leonhard, “Sur l’accord des deux dernieres eclipses du soleil et de lalune avec mes tables, pour trouver les vrais momens des pleni-lunes et novi-lunes,par M. Euler.” Memoires de l’academie des sciences de Berlin (1750). Reprintedin Opera Omnia: Series 2, Vol. 30, pp. 89–100. The original Latin and an Englishtranslation by Andrew Fabian are available at www.eulerarchive.org.

[E343] Euler, Leonhard, Letters of Euler on Different Subjects in Natural Philoso-phy Addressed to a German Princess, English translation with notes by DavidBrewster, Volume 1. Translation published by J. & J. Harper (1833). The Englishtranslation are available at www.eulerarchive.org.

[Euler 1743] Euler, Leonhard De causa gravitatis [publ. anonymously]. MiscellaneaBerolinensia, 7:360–370, 1743. Reprinted in Opera Omnia: Series II, Vol. 31, pp.

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373–378. The English translation by Ben DeSchmidt and Dominic Klyve is indraft form, and is available by writing the authors.

[Euler 1980] Euler, Leonhard, Opera Omnia, Series IVA, Volume 5. A.P. Juskevic, R.Taton, eds. Birkhauser (1980).

[Bradley 2007] Bradley, Robert, “Three Bodies? Why not Four? The Motion of theLunar Apsides,” published in Euler at 300 Mathematical Association of America(2007).

[Hankins 1985] Hankins, Thomas, Science and the Englightenment. Cambridge Uni-versity Press (1985).

[Kleinert 1996] Kleinert, Andreas, “Kommentar zu einer Schrift Eulers uber dieSchwerkraft”. Opera Omnia, Series II, Vol. 31 (1996), pp. LXXXVII – XCIII.

[Wilson 2007] Wilson, Curtis, “Euler and Applications of Analytical Mathematics toAstronomy,” published in Leonhard Euler: Life, Work, and Legacy (Bradley andSandifer, eds.), pp. 121–145 Elsevier (2007).

Ed Sandifer (SandiferE@wcsu.edu) is Professor of Mathematics at Western Connecti-cut State University in Danbury, CT. He is an avid runner, with 37 Boston Marathonson his shoes, and he is secretary of the Euler Society (www.EulerSociety.org). Hisfirst book, The Early Mathematics of Leonhard Euler, was published by the MAA inDecember 2006, as part of the celebration of Euler’s tercentennial in 2007. The MAApublished a collection of forty How Euler Did It columns in June 2007.

Dominic Klyve (dklyve@carthage.edu) is Assistant Professor of Mathematics at CarthageCollege in Kenosha, WI, and is a member of the Executive Board of the Euler Society.Dominic is also co-founder (with Lee Stemkoski) and director of the Euler Archive.Dominic has written several papers about Euler’s life and work, and has worked ontranslations of several of Euler’s papers.

How Euler Did It is updated each month.Copyright c©2009 Dominic Klyve and Ed Sandifer

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